How to set a numbered name for each row in a data frame?

You can do:

library(dplyr)

x %>%
  group_by(name) %>%
  mutate(new_name = paste(name, str_pad(row_number(), 2, pad = '0'), sep = '_'))

# A tibble: 6 × 3
# Groups:   name [3]
  name       id new_name
  <chr>   <dbl> <chr>   
1 a     2324545 a_01    
2 a      343245 a_02    
3 b       35435 b_01    
4 c      546565 c_01    
5 c       67432 c_02    
6 c       87865 c_03  

I would use aggregate() to calculate the "within-name ID number", paste0() to concatenate the strings, and formatC() to add leading zeroes.

x <- data.frame(
  name = c('a', 'a', 'b', 'c', 'c', 'c'),
  id = c(2324545, 343245, 35435, 546565, 67432, 87865)
)

x$new_name <- paste0(x$name, "_", 
                     formatC(
                       do.call('c', aggregate(rep(1, length(x$name)), 
                                              list(factor(x$name)), 
                                              FUN = cumsum)$x),
                       width = 2,
                       format = "d",
                       flag = "0"
                     ))
x
#>   name      id new_name
#> 1    a 2324545     a_01
#> 2    a  343245     a_02
#> 3    b   35435     b_01
#> 4    c  546565     c_01
#> 5    c   67432     c_02
#> 6    c   87865     c_03

With data.table

library(data.table)
setDT(x)[, new_name := sprintf('%s_%02d', name, rowid(name))]

-output

> x
     name      id new_name
   <char>   <num>    <char>
1:      a 2324545      a_01
2:      a  343245      a_02
3:      b   35435      b_01
4:      c  546565      c_01
5:      c   67432      c_02
6:      c   87865      c_03